What if the traditional relationship between touch and music was essentially turned upside down, making the tactile sensation the aesthetic end? This paper presents a novel coupling of haptics technology and music, introducing the notion of tactile composition or aesthetic composition for the sense of touch. A system that facilitates the composition and perception of intricate, musically structured spatio-temporal patterns of vibration on the surface of the body is described. Relevant work from disciplines including sensory substitution, electronic musical instrument design, simulation design, entertainment technology, and visual music is considered. The psychophysical parameter space for our sense of touch is summarized and the building blocks of a compositional language for touch are explored. A series of concerts held for the skin and ears is described, as well as some of the lessons learned along the way. In conclusion, some potential evolutionary branches of tactile composition are posited.